Specifically, I am trying to wrap my head around this scenario but can't do so solely by thinking about the weight of the fluid.
Imagine a perfectly still lake. If you dive into the lake, the further down you go, the more pressure you feel - presumably because of the weight of the water above you. Now, imagine that I replace some arbitrary chunk of water beneath the surface with a (rigid) solid object that is magically held in place. If I were to swim and place myself beneath this object, one would naively think that there is less weight of water above me (after all, we got rid of some chunk of fluid), but in fact I feel the same pressure on me regardless of whether the object is there or not. In essence, it's the height I am beneath the surface that matters, not the weight of the fluid above me. Why? What is generating this force if not the weight?